Explanation Given information: The radius of the disk is 1 m. Calculation: Let assume the cut made along a vertical line with x = b > 0 from the center of the disk. Sketch the disk with radius 1 m and the cut made at x = b as shown in Figure 1. Refer Figure 1. The disk boundary forms a circle. The equation of the circle is x 2 + y 2 = 1 (1) The center of mass of the small piece is ( 1 2 , 0 ) . To find the distance b the cut to be made that is closer to the center of the disk: Rearrange Equation (1) as shown below. y 2 = 1 − x 2 y = 1 − x 2 The expression for centroid of the uniform disk is shown below. x ¯ = ∫ a b x f ( x ) d x ∫ a b f ( x ) d x (2) Here, the curve function is f ( x ) , the lower limit is a , and the upper limit is b . Rearrange Equation (2) for the curve symmetry. x ¯ = ∫ a b 2 x f ( x ) d x ∫ a b 2 f ( x ) d x (3) Substitute 1 2 for x ¯ , 1 − x 2 for f ( x ) , b for a , and 1 for b in Equation (3). 1 2 = ∫ b 1 x × 2 1 − x 2 d x ∫ b 1 2 1 − x 2 d x 1 2 = − ∫ b 1 ( 1 − x 2 )

Single Variable Calculus: Early Transcendentals

8th Edition

ISBN: 9781305270336

Author: James Stewart

Publisher: Cengage Learning

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Chapter 8, Problem 9P

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Chapter 8, Problem 8P

Chapter 8, Problem 10P

Chapter 8 Solutions

Single Variable Calculus: Early Transcendentals

Ch. 8.1 - Use the arc length formula (3) to find the length...Ch. 8.1 - Prob. 2E Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Prob. 4E Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E Ch. 8.1 - Set up an integral that represents the length of...Ch. 8.1 - Find the exact length of the curve. 9. y = 1 +...Ch. 8.1 - Find the exact length of the curve. 10. 36y2 = (x2...

Ch. 8.1 - Find the exact length of the curve. 11....Ch. 8.1 - Find the exact length of the curve. 12....Ch. 8.1 - Prob. 13E Ch. 8.1 - Prob. 14E Ch. 8.1 - Prob. 15E Ch. 8.1 - Prob. 16E Ch. 8.1 - Prob. 17E Ch. 8.1 - Find the exact length of the curve. 18....Ch. 8.1 - Prob. 19E Ch. 8.1 - Prob. 20E Ch. 8.1 - Find the length of the arc of the curve from point...Ch. 8.1 - Prob. 22E Ch. 8.1 - Prob. 23E Ch. 8.1 - Prob. 24E Ch. 8.1 - Prob. 25E Ch. 8.1 - Prob. 26E Ch. 8.1 - Prob. 27E Ch. 8.1 - Prob. 28E Ch. 8.1 - Prob. 29E Ch. 8.1 - Prob. 30E Ch. 8.1 - Sketch the curve with equation x2/3 + y2/3 = 1 and...Ch. 8.1 - Prob. 34E Ch. 8.1 - Prob. 35E Ch. 8.1 - (a) Find the arc length function for the curve y =...Ch. 8.1 - Find the arc length function for the curve...Ch. 8.1 - The arc length function for a curve y = f(x),...Ch. 8.1 - Prob. 39E Ch. 8.1 - Prob. 40E Ch. 8.1 - A hawk flying at 15 m/s at an altitude of 180 m...Ch. 8.1 - Prob. 42E Ch. 8.1 - A manufacturer of corrugated metal roofing wants...Ch. 8.1 - (a) The figure shows a telephone wire hanging...Ch. 8.1 - Prob. 45E Ch. 8.1 - The curves with equations x + y = l , n = 4, 6, 8,...Ch. 8.2 - (a) Set up an integral for the area of the surface...Ch. 8.2 - (a) Set up an integral for the area of the surface...Ch. 8.2 - (a) Set up an integral for the area of the surface...Ch. 8.2 - (a) Set up an integral for the area of the surface...Ch. 8.2 - Prob. 5E Ch. 8.2 - (a) Set up an integral for the area of the surface...Ch. 8.2 - Prob. 7E Ch. 8.2 - Prob. 8E Ch. 8.2 - Prob. 9E Ch. 8.2 - Prob. 10E Ch. 8.2 - Prob. 11E Ch. 8.2 - Prob. 12E Ch. 8.2 - Prob. 13E Ch. 8.2 - Find the exact area of the surface obtained by...Ch. 8.2 - Prob. 15E Ch. 8.2 - The given curve is rotated about the y-axis. Find...Ch. 8.2 - Prob. 17E Ch. 8.2 - Prob. 18E Ch. 8.2 - Prob. 19E Ch. 8.2 - Prob. 20E Ch. 8.2 - Prob. 21E Ch. 8.2 - Prob. 22E Ch. 8.2 - Prob. 23E Ch. 8.2 - Prob. 24E Ch. 8.2 - Prob. 27E Ch. 8.2 - Prob. 28E Ch. 8.2 - Prob. 29E Ch. 8.2 - Prob. 30E Ch. 8.2 - Prob. 31E Ch. 8.2 - Prob. 32E Ch. 8.2 - If the curve y = f(x), a x b, is rotated about...Ch. 8.2 - Find the area of the surface obtained by rotating...Ch. 8.2 - (a) Show that the surface area of a zone of a...Ch. 8.2 - Prob. 37E Ch. 8.2 - Prob. 38E Ch. 8.2 - Formula 4 is valid only when f(x) 0. Show that...Ch. 8.3 - An aquarium 5 ft long, 2 ft wide, and 3 ft deep is...Ch. 8.3 - Prob. 2E Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - Prob. 5E Ch. 8.3 - Prob. 6E Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - A vertical plate is submerged (or partially...Ch. 8.3 - Prob. 11E Ch. 8.3 - Prob. 12E Ch. 8.3 - A trough is filled with a liquid of density 840...Ch. 8.3 - A vertical dam has a semicircular gate as shown in...Ch. 8.3 - Prob. 15E Ch. 8.3 - Prob. 16E Ch. 8.3 - A swimming pool is 20 ft wide and 40 ft long and...Ch. 8.3 - Prob. 18E Ch. 8.3 - Prob. 19E Ch. 8.3 - Prob. 20E Ch. 8.3 - Point-masses mi are located on the x-axis as...Ch. 8.3 - Prob. 22E Ch. 8.3 - Prob. 23E Ch. 8.3 - Prob. 24E Ch. 8.3 - Prob. 25E Ch. 8.3 - Prob. 26E Ch. 8.3 - Prob. 27E Ch. 8.3 - Prob. 28E Ch. 8.3 - Prob. 29E Ch. 8.3 - Prob. 30E Ch. 8.3 - Prob. 31E Ch. 8.3 - Prob. 32E Ch. 8.3 - Find the centroid of the region bounded by the...Ch. 8.3 - Calculate the moments Mx and My and the center of...Ch. 8.3 - Calculate the moments Mx and My and the center of...Ch. 8.3 - Prob. 36E Ch. 8.3 - Prob. 37E Ch. 8.3 - Prob. 38E Ch. 8.3 - Prob. 39E Ch. 8.3 - Prob. 40E Ch. 8.3 - Prob. 41E Ch. 8.3 - Prob. 42E Ch. 8.3 - Prob. 43E Ch. 8.3 - Use the Theorem of Pappus to find the volume of...Ch. 8.3 - Prob. 45E Ch. 8.3 - Prob. 46E Ch. 8.3 - Prob. 47E Ch. 8.3 - Prob. 48E Ch. 8.3 - Use the Second Theorem of Pappus described in...Ch. 8.3 - Prob. 50E Ch. 8.3 - Prob. 51E Ch. 8.4 - Prob. 1E Ch. 8.4 - Prob. 2E Ch. 8.4 - Prob. 3E Ch. 8.4 - Prob. 4E Ch. 8.4 - Prob. 5E Ch. 8.4 - Prob. 6E Ch. 8.4 - If a supply curve is modeled by the equation p =...Ch. 8.4 - Prob. 8E Ch. 8.4 - Prob. 9E Ch. 8.4 - Prob. 10E Ch. 8.4 - Prob. 11E Ch. 8.4 - Prob. 12E Ch. 8.4 - Prob. 13E Ch. 8.4 - Prob. 14E Ch. 8.4 - Prob. 15E Ch. 8.4 - Prob. 16E Ch. 8.4 - Prob. 17E Ch. 8.4 - Prob. 18E Ch. 8.4 - Prob. 19E Ch. 8.4 - Prob. 20E Ch. 8.4 - Prob. 21E Ch. 8.4 - Prob. 22E Ch. 8.4 - Prob. 23E Ch. 8.5 - Prob. 1E Ch. 8.5 - Prob. 2E Ch. 8.5 - Prob. 3E Ch. 8.5 - Prob. 4E Ch. 8.5 - Prob. 5E Ch. 8.5 - Let f(x) = k (3x x2) if 0 x 3 and f(x) = 0 if x...Ch. 8.5 - Prob. 7E Ch. 8.5 - Prob. 8E Ch. 8.5 - Prob. 9E Ch. 8.5 - Prob. 10E Ch. 8.5 - Prob. 11E Ch. 8.5 - Prob. 12E Ch. 8.5 - REM sleep is the phase of sleep when most active...Ch. 8.5 - Prob. 14E Ch. 8.5 - The Garbage Project at the University of Arizona...Ch. 8.5 - Prob. 16E Ch. 8.5 - The speeds of vehicles on a highway with speed...Ch. 8.5 - Prob. 18E Ch. 8.5 - Prob. 19E Ch. 8.5 - The standard deviation for a random variable with...Ch. 8.5 - Prob. 21E Ch. 8 - (a) How is the length of a curve defined? (b)...Ch. 8 - Prob. 2RCC Ch. 8 - Describe how we can find the hydrostatic force...Ch. 8 - (a) What is the physical significance of the...Ch. 8 - Prob. 5RCC Ch. 8 - Prob. 6RCC Ch. 8 - Prob. 7RCC Ch. 8 - Prob. 8RCC Ch. 8 - Prob. 9RCC Ch. 8 - Prob. 10RCC Ch. 8 - Prob. 1RE Ch. 8 - Prob. 2RE Ch. 8 - Prob. 3RE Ch. 8 - Prob. 4RE Ch. 8 - Prob. 5RE Ch. 8 - Prob. 6RE Ch. 8 - Prob. 7RE Ch. 8 - Prob. 8RE Ch. 8 - Prob. 9RE Ch. 8 - Prob. 10RE Ch. 8 - A gate in an irrigation canal is constructed in...Ch. 8 - A trough is filled with water and its vertical...Ch. 8 - Find the centroid of the region shown. 13.Ch. 8 - Prob. 14RE Ch. 8 - Prob. 15RE Ch. 8 - Prob. 16RE Ch. 8 - Prob. 17RE Ch. 8 - Prob. 18RE Ch. 8 - Prob. 19RE Ch. 8 - Prob. 20RE Ch. 8 - Prob. 21RE Ch. 8 - Prob. 22RE Ch. 8 - Prob. 23RE Ch. 8 - Prob. 1P Ch. 8 - Prob. 2P Ch. 8 - Prob. 3P Ch. 8 - (a) Show that an observer at height H above the...Ch. 8 - Prob. 5P Ch. 8 - Prob. 6P Ch. 8 - Prob. 7P Ch. 8 - Prob. 8P Ch. 8 - Prob. 9P Ch. 8 - Prob. 10P Ch. 8 - Prob. 11P Ch. 8 - Prob. 12P Ch. 8 - Prob. 13P

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The distance the cut to be made from the center of the disk.

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Chapter 8 Solutions